Abstract
We numerically investigate triple collision orbits of the free-fall three-body system which has no double collisions before three bodies collide. Triple collision is an important property of the three-body system. Tanikawa, Saito, Mikkola (Celest Mech Dyn Astron 131(6):24, 2019) obtained 11 triple collision orbits without double collision for the free-fall three-body problem. In this paper, we present 1658 triple collision orbits including the Lagrange’s homothetic solution, 11 ones found by Tanikawa et al. (2019) and 1646 new triple collision orbits. The symbol sequences of these 1646 new triple collision orbits have digits that range between 1 and 120. With our high-precision results, numerical evidences of the asymptotic property of triple collision orbits are given.
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References
Abad, A., Barrio, R., Dena, A.: Computing periodic orbits with arbitrary precision. Phys. Rev. E 84, 016701 (2011)
Agekyan, T.A., Anosova, Z.P.: A study of the dynamics of triple systems by means of statistical sampling. Soviet Phys. Astron. 11, 1006 (1968)
Barrio, R., Blesa, F., Lara, M.: VSVO formulation of the Taylor method for the numerical solution of ODEs. Comput. Math. Appl. 50(1), 93–111 (2005)
Barton, D., Willem, I., Zahar, R.: The automatic solution of ordinary differential equations by the method of Taylor series. Comput. J. 14, 243–248 (1971)
Belbruno, E., Frauenfelder, U., van Koert, O.: A family of periodic orbits in the three-dimensional lunar problem. Celest. Mech. Dyn. Astron. 131(2), 7 (2019)
Burrau, C.: Numerische berechnung eines spezialfalles des dreikörperproblems. Astron. Nachr. 195, 113 (1913)
Dmitrašinović, V., Šuvakov, M.: A guide to hunting periodic three-body orbits. Am. J. Phys. 82(6), 609–619 (2014)
Chang, Y.F., Corhss, G.F.: ATOMFT: solving ODEs and DAEs using Taylor series. Comput. Math. Appl. 28, 209–233 (1994)
Chen, N.C.: Periodic brake orbits in the planar isosceles three-body problem. Nonlinearity 26(10), 2875 (2013)
Corliss, G., Chang, Y.: Solving ordinary differential equations using Taylor series. ACM Trans. Math. Softw. 8, 114–144 (1982)
Devaney, R.L.: Triple collision in the planar isosceles three body problem. Invent. Math. 60(3), 249–267 (1980)
Farantos, S.C.: Methods for locating periodic orbits in highly unstable systems. J. Mol. Struct. (Thoechem) 341(1), 91–100 (1995)
Fousse, L., Hanrot, G., Lefèvre, V., Pélissier, P., Zimmermann, P.: Mpfr: a multiple-precision binary floating-point library with correct rounding. ACM Trans. Math. Softw. (TOMS) 33(2), 13 (2007)
Gao, F., Llibre, J.: Periodic orbits of the two fixed centers problem with a variational gravitational field. Celest. Mech. Dyn. Astron. 132(6), 1–9 (2020)
Hairer, E., Wanner, G., Norsett, S.P.: Solving Ordinary Differential Equations I: Non-stiff Problems. Springer, Berlin (1993)
He, M.Y., Petrovich, C.: On the stability and collisions in triple stellar systems. Mon. Not. R. Astron. Soc. 474(1), 20–31 (2018)
Hu, T., Liao, S.: On the risks of using double precision in numerical simulations of spatio-temporal chaos. J. Comput. Phys. 418, 109629 (2020)
Iasko, P.P., Orlov, V.V.: Search for periodic orbits in the general three-body problem. Astron. Rep. 58(11), 869–879 (2014)
Lara, M., Pelaez, J.: On the numerical continuation of periodic orbits-an intrinsic, 3-dimensional, differential, predictor-corrector algorithm. Astron. Astrophys. 389(2), 692–701 (2002)
Li, X., Liao, S.: More than six hundred new families of newtonian periodic planar collisionless three-body orbits. Sci. China Phys. Mech. Astron. 60(12), 129511 (2017)
Li, X., Liao, S.: Collisionless periodic orbits in the free-fall three-body problem. New Astron. 70, 22–26 (2019)
Li, X., Jing, Y., Liao, S.: Over a thousand new periodic orbits of a planar three-body system with unequal masses. Publ. Astron. Soc. Jpn. 70(4), 64 (2018)
Li, X., Li, X., Liao, S.: One family of 13315 stable periodic orbits of non-hierarchical unequal-mass triple systems. Sci. China Phys. Mech. Astron. 64(1), 1–6 (2021)
Liao, S.: On the reliability of computed chaotic solutions of non-linear differential equations. Tellus A 61(4), 550–564 (2009)
Liao, S.: Physical limit of prediction for chaotic motion of three-body problem. Commun. Nonlinear Sci. Numer. Simul. 19(3), 601–616 (2014)
Liao, S., Wang, P.: On the mathematically reliable long-term simulation of chaotic solutions of lorenz equation in the interval [0,10000]. Sci. China Phys. Mech. Astron. 57, 330–335 (2014)
McGehee, R.: Triple collision in the collinear three-body problem. Invent. Math. 27(3), 191–227 (1974)
Montgomery, R.: The n -body problem, the braid group, and action-minimizing periodic solutions. Nonlinearity 11(2), 363 (1998)
Montgomery, R.: The zero angular momentum, three-body problem: All but one solution has syzygies. Ergodic Theory Dyn. Syst. 27(6), 1933–1946 (2007)
Standish, E.: New periodic orbits in the general problem of three bodies. In: Giacaglia, G.E.O. (ed.) Periodic Orbits, Stability and Resonances. Springer, Dordrecht (1970)
Stone, N.C., Leigh, N.W.: A statistical solution to the chaotic, non-hierarchical three-body problem. Nature 576(7787), 406–410 (2019)
Sundman, K.: Nouvelles recherches sur le probléme des trois corps. Acta Soc. Sci. Fenn 35, 9 (1909)
Szebehely, V., Peters, C.F.: A new periodic solution of the problem of three bodies. Astron. J. 3, 17 (1967)
Tanikawa, K.: A search for collision orbits in the free-fall three-body problem II. Celest. Mech. Dyn. Astron. 76(3), 157–185 (2000)
Tanikawa, K., Mikkola, S.: Symbol sequences and orbits of the free-fall three-body problem. Publ. Astron. Soc. Jpn. 67(6), 806 (2015)
Tanikawa, K., Umehara, H., Abe, H.: A search for collision orbits in the free-fall three-body problem i. Numerical procedure. Celest. Mech. Dyn. Astron. 62(4), 335–362 (1995)
Tanikawa, K., Saito, M.M., Mikkola, S.: A search for triple collision orbits inside the domain of the free-fall three-body problem. Celest. Mech. Dyn. Astron. 131(6), 24 (2019)
Trefethen, L., Bau, D., III.: Numerical Linear Algebra. Society for Industrial and Applied Mathematics, Philadelphia, PA (1997)
Umehara, H., Tanikawa, K.: Binary and triple collisions causing instability in the free-fall three-body problem. Celest. Mech. Dyn. Astron. 76(3), 187–214 (2000)
Urminsky, D.J., Heggie, D.C.: On the relationship between instability and lyapunov times for the three-body problem. Mon. Not. R. Astron. Soc. 392(3), 1051–1059 (2010)
Šuvakov, M., Dmitrašinović, V.: Three classes of newtonian three-body planar periodic orbits. Phys. Rev. Lett. 110, 114301 (2013)
Yasko, P.P., Orlov, V.V.: Search for periodic orbits in agekyan and anosova’s region d for the general three-body problem. Astron. Rep. 59(5), 404–413 (2015)
Acknowledgements
This work was carried out on TH-1A at National Supercomputer Center in Tianjin, China. It is partly supported by National Natural Science Foundation of China (Approval Nos. 12002132, 11702099 and 91752104).
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Li, X., Li, X., He, L. et al. Triple collision orbits in the free-fall three-body system without binary collisions. Celest Mech Dyn Astr 133, 46 (2021). https://doi.org/10.1007/s10569-021-10044-6
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DOI: https://doi.org/10.1007/s10569-021-10044-6